cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

User: Lyle Ramshaw

Lyle Ramshaw's wiki page.

Lyle Ramshaw has authored 5 sequences.

A137400 a(n) = A128941(n) + 2.

Original entry on oeis.org

6, 30, 140, 631, 2786, 12136, 52368, 224406, 956516, 4060038, 17175132, 72454075, 304941386, 1280898304, 5371301504, 22491017758, 94055344244, 392888085100, 1639534704632, 6835739258998, 28477594607348, 118551827347576
Offset: 0

Author

Lyle Ramshaw (lyle.ramshaw(AT)hp.com), Apr 08 2008

Keywords

Crossrefs

Cf. A128941.

A128941 Cardinality of the free modular lattice generated by two elements and a chain of length n.

Original entry on oeis.org

4, 28, 138, 629, 2784, 12134, 52366, 224404, 956514, 4060036, 17175130, 72454073, 304941384, 1280898302, 5371301502, 22491017756, 94055344242, 392888085098, 1639534704630, 6835739258996, 28477594607346, 118551827347574
Offset: 0

Author

Lyle Ramshaw (lyle.ramshaw(AT)hp.com), Apr 08 2008

Keywords

Comments

If you choose to adjoin a top and a bottom element to each resulting lattice, you must add 2 to these cardinalities: see A137400.

Examples

			When n = 0, the lattice consists of the two elements, their meet and their join, so a(0) = 4.
When n = 1, we get the free modular lattice generated by three elements, so a(1) = 28.
		

References

  • G. Birkoff, Lattice Theory, American Mathematical Society, third edition (1967), pp. 63-64 [for the case n = 1].

Crossrefs

Cf. A137400.

Extensions

More terms from Vladeta Jovovic, Feb 05 2010

A023813 a(n) = n^(n*(n+1)/2).

Original entry on oeis.org

1, 1, 8, 729, 1048576, 30517578125, 21936950640377856, 459986536544739960976801, 324518553658426726783156020576256, 8727963568087712425891397479476727340041449, 10000000000000000000000000000000000000000000000000000000
Offset: 0

Author

Lyle Ramshaw (ramshaw(AT)pa.dec.com)

Keywords

Comments

Determinant of n X n matrix M_(i,j) = binomial(n*i,j). - Benoit Cloitre, Sep 13 2003
Number of commutative binary operations on an n-set. Labeled commutative groupoids.

Crossrefs

Programs

Formula

a(n) = Product_{k=1..n} n^k. - José de Jesús Camacho Medina, Jul 12 2016
a(n) = n^A000217(n). - Alois P. Heinz, Aug 06 2018

Extensions

Better description from Amarnath Murthy, Dec 29 2001

A023814 Number of associative binary operations on an n-set; number of labeled semigroups.

Original entry on oeis.org

1, 1, 8, 113, 3492, 183732, 17061118, 7743056064, 148195347518186, 38447365355811944462
Offset: 0

Author

Lyle Ramshaw (ramshaw(AT)pa.dec.com)

Keywords

Crossrefs

Extensions

a(8), a(9) from Distler and Kelsey (2013). - N. J. A. Sloane, Feb 19 2013

A023815 Number of binary operations on an n-set that are commutative and associative; labeled commutative semigroups.

Original entry on oeis.org

1, 1, 6, 63, 1140, 30730, 1185072, 66363206, 7150843144, 3829117403448
Offset: 0

Author

Lyle Ramshaw (ramshaw(AT)pa.dec.com)

Keywords

Crossrefs

Row sums of A058167.
Cf. A001423, A001426 (isomorphism classes), A023813 (commutative only), A023814 (associative only), A027851.

Formula

a(n) + A079192(n) + A079195(n) + A079198(n) = A002489(n).
a(n) = Sum_{k>=1} A079201(n,k)*A079210(n,k). - Andrew Howroyd, Jan 26 2022

Extensions

a(8) from Andrew Howroyd, Jan 26 2022
a(9) from Andrew Howroyd, Feb 14 2022